トムソン問題

トムソン問題は、数学者スティーヴン・スメイルによる18個の未解決問題の1つ — "2次元球面上の点の分布" — の特別な場合である[2]。冒頭述べたように単位球面（ ${\displaystyle r=1}$ ）の場合だけ考えればよい。

電気量 ${\displaystyle e_{i}=e_{j}=e}$ （ ${\displaystyle e}$ は電気素量）の電子ペアがもつ静電エネルギーは、クーロンの法則より

${\displaystyle U_{ij}(N)=k_{e}{e_{i}e_{j} \over r_{ij}}}$

ここで ${\displaystyle k_{e}}$ はクーロン定数、 ${\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}$ は位置ベクトル ${\displaystyle \mathbf {r} _{i}}$、${\displaystyle \mathbf {r} _{j}}$ で表した電子間距離。

${\displaystyle e=1}$、${\displaystyle k_{e}=1}$ であるとしても一般性は失わない。このとき、

${\displaystyle U_{ij}(N)={1 \over r_{ij}}}$

N-電子配置のエネルギーの総和は、ペアごとの値を合算して次のように書ける。

${\displaystyle U(N)=\sum _{i<j}{\frac {1}{r_{ij}}}}$

大局的な ${\displaystyle U(N)}$ の最小化配置は、大体において数値計算で得られる。

N=5までのトムソン問題の解を図示したもの

最小エネルギー配置が厳密に分かっているのはほんの限られた場合だけである。

• N =1のとき：解は自明であり、電子は球面上のどこにあってもよい。このときのエネルギーは、電子が別の電場源からの影響を一切受けないことからゼロと定義することにする。
• N =2のとき：2電子が対蹠点に位置するときが解。
• N =3のとき：3電子が大円上の正三角形の頂点になるときが解[3]。
• N =4のとき：4電子が正四面体の頂点になるときが解。
• N =5のとき：コンピュータを用いた数学的厳密解が2010年に報告された。5電子が双三角錐の頂点になるときが解[4]。
• N =6のとき：6電子が正八面体の頂点になるときが解[5]。
• N =12のとき：12電子が正二十面体の頂点になるときが解[6]。

注目すべきことに、 N =4,6,12のときのトムソン問題の解は、全ての面が合同な正三角形であるプラトンの立体を形作っている。 N =8,20のときの数値計算による解は残り2つのプラトンの立体（面が正方形、正五角形）にはなっていない。

粒子間のポテンシャルが任意の形で与えられているときの基底状態を考えることもできる。数学的に正式に述べると、 f を実数値減少関数とし、全エネルギーを ${\displaystyle \sum _{i<j}f(|\mathbf {r} _{i}-\mathbf {r} _{j}|)}$ とする。

伝統的に ${\displaystyle f(x)=x^{-\alpha }}$ （リースの ${\displaystyle \alpha }$-カーネル）を考えることが多い。可積分なリースカーネルについては文献[7]を参照のこと。また可積分でないリースカーネルについてはPoppy-seed bagel theoremが成り立っている[8]。

有名なケースは

• α → ∞ としたとき：「球面上の2点間の距離の最大値が最小となる配置を求める問題」と解釈する。Tammes problemとして知られている。
• α = 1のとき：トムソン問題。
• α = 0のとき：${\displaystyle f(x)=\ln \left({\frac {1}{|x|}}\right)}$ と解釈する。Whyte's problemとして知られている。

高次元空間での N 個の点の配置を考えることもできる。球デザインを参照。

トムソン問題は、トムソンのブドウパンモデルにおいて一様に広がる正電荷がないものとすれば自然に導かれる[9]。

実験によってトムソンのブドウパンモデルは原子の完全なモデルにはなり得ないことがわかったが、数値計算によるトムソン問題の変則的な解の中には、周期表上の一連の元素における自然な殻内電子配置と対応が見つかっているものもある[11]。

トムソン問題はまた、電子バブルや、四重極イオントラップ内の金属液滴の表面配向といった他の物理モデルの研究でも有用である。

一般化されたトムソン問題は、例えばウイルスの球状殻を構成するタンパク質のサブユニット配置を決定するときに現れる。この場合の「粒子」とは殻上のサブユニットのことである。他の具体例として、コロイドソーム（colloidosome；薬物、栄養物、生細胞等、活性成分のカプセル封じとして提案されている）におけるコロイド粒子の均整な配置、炭素原子のフラーレン構造、原子価殻電子対反発則の理論等が挙げられる。低温状態で、巨大単極子を中心としたシェル構造超伝導金属が形成するとされるアブリコソフ-ボルテックスは、対数ポテンシャルによる長距離相互作用の例である。

下記の表で、${\displaystyle N}$ は点（電荷）の個数、 ${\displaystyle E_{1}}$ はエネルギー、対称性の記述はシェーンフリース記号（三次元の点群を参照）によるもの、また ${\displaystyle r_{i}}$ は点電荷の位置を表す。ほとんどの対称性のタイプにおいてベクトルの総和はゼロになり、従って電気双極子モーメントもゼロになる。

点集合の凸包による多面体も併せて考えることが慣習となっている。ここで、${\displaystyle v_{i}}$ は添字に等しい辺が集まる頂点の数、${\displaystyle e}$ は辺の数、${\displaystyle f_{3}}$ は三角形の面の数、${\displaystyle f_{4}}$ は四角形の面の数、${\displaystyle \theta _{1}}$ は隣接する2辺が成す角の最小値である。辺の長さは一定とは限らないことに注意。このため（ N =4, 6, 12, 24を除いて） 凸包は表の最終列に記した一様多面体またはジョンソンの立体とはグラフ同型であるにとどまる[12]。

N	${\displaystyle E_{1}}$	対称性	${\displaystyle \left|\sum \mathbf {r} _{i}\right|}$	${\displaystyle v_{3}}$	${\displaystyle v_{4}}$	${\displaystyle v_{5}}$	${\displaystyle v_{6}}$	${\displaystyle v_{7}}$	${\displaystyle v_{8}}$	${\displaystyle e}$	${\displaystyle f_{3}}$	${\displaystyle f_{4}}$	${\displaystyle \theta _{1}}$	対応する多面体
2	0.500000000	${\displaystyle D_{\infty h}}$	0	–	–	–	–	–	–	1	–	–	180.000°	二角形
3	1.732050808	${\displaystyle D_{3h}}$	0	–	–	–	–	–	–	3	1	–	120.000°	三角形
4	3.674234614	${\displaystyle T_{d}}$	0	4	0	0	0	0	0	6	4	0	109.471°	四面体
5	6.474691495	${\displaystyle D_{3h}}$	0	2	3	0	0	0	0	9	6	0	90.000°	双三角錐
6	9.985281374	${\displaystyle O_{h}}$	0	0	6	0	0	0	0	12	8	0	90.000°	八面体
7	14.452977414	${\displaystyle D_{5h}}$	0	0	5	2	0	0	0	15	10	0	72.000°	双五角錐
8	19.675287861	${\displaystyle D_{4d}}$	0	0	8	0	0	0	0	16	8	2	71.694°	反四角柱形
9	25.759986531	${\displaystyle D_{3h}}$	0	0	3	6	0	0	0	21	14	0	69.190°	三側錐三角柱
10	32.716949460	${\displaystyle D_{4d}}$	0	0	2	8	0	0	0	24	16	0	64.996°	双四角錐反柱
11	40.596450510	${\displaystyle C_{2v}}$	0.013219635	0	2	8	1	0	0	27	18	0	58.540°	edge-contracted icosahedron
12	49.165253058	${\displaystyle I_{h}}$	0	0	0	12	0	0	0	30	20	0	63.435°	正二十面体
13	58.853230612	${\displaystyle C_{2v}}$	0.008820367	0	1	10	2	0	0	33	22	0	52.317°
14	69.306363297	${\displaystyle D_{6d}}$	0	0	0	12	2	0	0	36	24	0	52.866°	gyroelongated hexagonal dipyramid
15	80.670244114	${\displaystyle D_{3}}$	0	0	0	12	3	0	0	39	26	0	49.225°
16	92.911655302	${\displaystyle T}$	0	0	0	12	4	0	0	42	28	0	48.936°
17	106.050404829	${\displaystyle D_{5h}}$	0	0	0	12	5	0	0	45	30	0	50.108°
18	120.084467447	${\displaystyle D_{4d}}$	0	0	2	8	8	0	0	48	32	0	47.534°
19	135.089467557	${\displaystyle C_{2v}}$	0.000135163	0	0	14	5	0	0	50	32	1	44.910°
20	150.881568334	${\displaystyle D_{3h}}$	0	0	0	12	8	0	0	54	36	0	46.093°
21	167.641622399	${\displaystyle C_{2v}}$	0.001406124	0	1	10	10	0	0	57	38	0	44.321°
22	185.287536149	${\displaystyle T_{d}}$	0	0	0	12	10	0	0	60	40	0	43.302°
23	203.930190663	${\displaystyle D_{3}}$	0	0	0	12	11	0	0	63	42	0	41.481°
24	223.347074052	${\displaystyle O}$	0	0	0	24	0	0	0	60	32	6	42.065°	変形立方体
25	243.812760299	${\displaystyle C_{s}}$	0.001021305	0	0	14	11	0	0	68	44	1	39.610°
26	265.133326317	${\displaystyle C_{2}}$	0.001919065	0	0	12	14	0	0	72	48	0	38.842°
27	287.302615033	${\displaystyle D_{5h}}$	0	0	0	12	15	0	0	75	50	0	39.940°
28	310.491542358	${\displaystyle T}$	0	0	0	12	16	0	0	78	52	0	37.824°
29	334.634439920	${\displaystyle D_{3}}$	0	0	0	12	17	0	0	81	54	0	36.391°
30	359.603945904	${\displaystyle D_{2}}$	0	0	0	12	18	0	0	84	56	0	36.942°
31	385.530838063	${\displaystyle C_{3v}}$	0.003204712	0	0	12	19	0	0	87	58	0	36.373°
32	412.261274651	${\displaystyle I_{h}}$	0	0	0	12	20	0	0	90	60	0	37.377°
33	440.204057448	${\displaystyle C_{s}}$	0.004356481	0	0	15	17	1	0	92	60	1	33.700°
34	468.904853281	${\displaystyle D_{2}}$	0	0	0	12	22	0	0	96	64	0	33.273°
35	498.569872491	${\displaystyle C_{2}}$	0.000419208	0	0	12	23	0	0	99	66	0	33.100°
36	529.122408375	${\displaystyle D_{2}}$	0	0	0	12	24	0	0	102	68	0	33.229°
37	560.618887731	${\displaystyle D_{5h}}$	0	0	0	12	25	0	0	105	70	0	32.332°
38	593.038503566	${\displaystyle D_{6d}}$	0	0	0	12	26	0	0	108	72	0	33.236°
39	626.389009017	${\displaystyle D_{3h}}$	0	0	0	12	27	0	0	111	74	0	32.053°
40	660.675278835	${\displaystyle T_{d}}$	0	0	0	12	28	0	0	114	76	0	31.916°
41	695.916744342	${\displaystyle D_{3h}}$	0	0	0	12	29	0	0	117	78	0	31.528°
42	732.078107544	${\displaystyle D_{5h}}$	0	0	0	12	30	0	0	120	80	0	31.245°
43	769.190846459	${\displaystyle C_{2v}}$	0.000399668	0	0	12	31	0	0	123	82	0	30.867°
44	807.174263085	${\displaystyle O_{h}}$	0	0	0	24	20	0	0	120	72	6	31.258°
45	846.188401061	${\displaystyle D_{3}}$	0	0	0	12	33	0	0	129	86	0	30.207°
46	886.167113639	${\displaystyle T}$	0	0	0	12	34	0	0	132	88	0	29.790°
47	927.059270680	${\displaystyle C_{s}}$	0.002482914	0	0	14	33	0	0	134	88	1	28.787°
48	968.713455344	${\displaystyle O}$	0	0	0	24	24	0	0	132	80	6	29.690°
49	1011.557182654	${\displaystyle C_{3}}$	0.001529341	0	0	12	37	0	0	141	94	0	28.387°
50	1055.182314726	${\displaystyle D_{6d}}$	0	0	0	12	38	0	0	144	96	0	29.231°
51	1099.819290319	${\displaystyle D_{3}}$	0	0	0	12	39	0	0	147	98	0	28.165°
52	1145.418964319	${\displaystyle C_{3}}$	0.000457327	0	0	12	40	0	0	150	100	0	27.670°
53	1191.922290416	${\displaystyle C_{2v}}$	0.000278469	0	0	18	35	0	0	150	96	3	27.137°
54	1239.361474729	${\displaystyle C_{2}}$	0.000137870	0	0	12	42	0	0	156	104	0	27.030°
55	1287.772720783	${\displaystyle C_{2}}$	0.000391696	0	0	12	43	0	0	159	106	0	26.615°
56	1337.094945276	${\displaystyle D_{2}}$	0	0	0	12	44	0	0	162	108	0	26.683°
57	1387.383229253	${\displaystyle D_{3}}$	0	0	0	12	45	0	0	165	110	0	26.702°
58	1438.618250640	${\displaystyle D_{2}}$	0	0	0	12	46	0	0	168	112	0	26.155°
59	1490.773335279	${\displaystyle C_{2}}$	0.000154286	0	0	14	43	2	0	171	114	0	26.170°
60	1543.830400976	${\displaystyle D_{3}}$	0	0	0	12	48	0	0	174	116	0	25.958°
61	1597.941830199	${\displaystyle C_{1}}$	0.001091717	0	0	12	49	0	0	177	118	0	25.392°
62	1652.909409898	${\displaystyle D_{5}}$	0	0	0	12	50	0	0	180	120	0	25.880°
63	1708.879681503	${\displaystyle D_{3}}$	0	0	0	12	51	0	0	183	122	0	25.257°
64	1765.802577927	${\displaystyle D_{2}}$	0	0	0	12	52	0	0	186	124	0	24.920°
65	1823.667960264	${\displaystyle C_{2}}$	0.000399515	0	0	12	53	0	0	189	126	0	24.527°
66	1882.441525304	${\displaystyle C_{2}}$	0.000776245	0	0	12	54	0	0	192	128	0	24.765°
67	1942.122700406	${\displaystyle D_{5}}$	0	0	0	12	55	0	0	195	130	0	24.727°
68	2002.874701749	${\displaystyle D_{2}}$	0	0	0	12	56	0	0	198	132	0	24.433°
69	2064.533483235	${\displaystyle D_{3}}$	0	0	0	12	57	0	0	201	134	0	24.137°
70	2127.100901551	${\displaystyle D_{2d}}$	0	0	0	12	50	0	0	200	128	4	24.291°
71	2190.649906425	${\displaystyle C_{2}}$	0.001256769	0	0	14	55	2	0	207	138	0	23.803°
72	2255.001190975	${\displaystyle I}$	0	0	0	12	60	0	0	210	140	0	24.492°
73	2320.633883745	${\displaystyle C_{2}}$	0.001572959	0	0	12	61	0	0	213	142	0	22.810°
74	2387.072981838	${\displaystyle C_{2}}$	0.000641539	0	0	12	62	0	0	216	144	0	22.966°
75	2454.369689040	${\displaystyle D_{3}}$	0	0	0	12	63	0	0	219	146	0	22.736°
76	2522.674871841	${\displaystyle C_{2}}$	0.000943474	0	0	12	64	0	0	222	148	0	22.886°
77	2591.850152354	${\displaystyle D_{5}}$	0	0	0	12	65	0	0	225	150	0	23.286°
78	2662.046474566	${\displaystyle T_{h}}$	0	0	0	12	66	0	0	228	152	0	23.426°
79	2733.248357479	${\displaystyle C_{s}}$	0.000702921	0	0	12	63	1	0	230	152	1	22.636°
80	2805.355875981	${\displaystyle D_{4d}}$	0	0	0	16	64	0	0	232	152	2	22.778°
81	2878.522829664	${\displaystyle C_{2}}$	0.000194289	0	0	12	69	0	0	237	158	0	21.892°
82	2952.569675286	${\displaystyle D_{2}}$	0	0	0	12	70	0	0	240	160	0	22.206°
83	3027.528488921	${\displaystyle C_{2}}$	0.000339815	0	0	14	67	2	0	243	162	0	21.646°
84	3103.465124431	${\displaystyle C_{2}}$	0.000401973	0	0	12	72	0	0	246	164	0	21.513°
85	3180.361442939	${\displaystyle C_{2}}$	0.000416581	0	0	12	73	0	0	249	166	0	21.498°
86	3258.211605713	${\displaystyle C_{2}}$	0.001378932	0	0	12	74	0	0	252	168	0	21.522°
87	3337.000750014	${\displaystyle C_{2}}$	0.000754863	0	0	12	75	0	0	255	170	0	21.456°
88	3416.720196758	${\displaystyle D_{2}}$	0	0	0	12	76	0	0	258	172	0	21.486°
89	3497.439018625	${\displaystyle C_{2}}$	0.000070891	0	0	12	77	0	0	261	174	0	21.182°
90	3579.091222723	${\displaystyle D_{3}}$	0	0	0	12	78	0	0	264	176	0	21.230°
91	3661.713699320	${\displaystyle C_{2}}$	0.000033221	0	0	12	79	0	0	267	178	0	21.105°
92	3745.291636241	${\displaystyle D_{2}}$	0	0	0	12	80	0	0	270	180	0	21.026°
93	3829.844338421	${\displaystyle C_{2}}$	0.000213246	0	0	12	81	0	0	273	182	0	20.751°
94	3915.309269620	${\displaystyle D_{2}}$	0	0	0	12	82	0	0	276	184	0	20.952°
95	4001.771675565	${\displaystyle C_{2}}$	0.000116638	0	0	12	83	0	0	279	186	0	20.711°
96	4089.154010060	${\displaystyle C_{2}}$	0.000036310	0	0	12	84	0	0	282	188	0	20.687°
97	4177.533599622	${\displaystyle C_{2}}$	0.000096437	0	0	12	85	0	0	285	190	0	20.450°
98	4266.822464156	${\displaystyle C_{2}}$	0.000112916	0	0	12	86	0	0	288	192	0	20.422°
99	4357.139163132	${\displaystyle C_{2}}$	0.000156508	0	0	12	87	0	0	291	194	0	20.284°
100	4448.350634331	${\displaystyle T}$	0	0	0	12	88	0	0	294	196	0	20.297°
101	4540.590051694	${\displaystyle D_{3}}$	0	0	0	12	89	0	0	297	198	0	20.011°
102	4633.736565899	${\displaystyle D_{3}}$	0	0	0	12	90	0	0	300	200	0	20.040°
103	4727.836616833	${\displaystyle C_{2}}$	0.000201245	0	0	12	91	0	0	303	202	0	19.907°
104	4822.876522746	${\displaystyle D_{6}}$	0	0	0	12	92	0	0	306	204	0	19.957°
105	4919.000637616	${\displaystyle D_{3}}$	0	0	0	12	93	0	0	309	206	0	19.842°
106	5015.984595705	${\displaystyle D_{2}}$	0	0	0	12	94	0	0	312	208	0	19.658°
107	5113.953547724	${\displaystyle C_{2}}$	0.000064137	0	0	12	95	0	0	315	210	0	19.327°
108	5212.813507831	${\displaystyle C_{2}}$	0.000432525	0	0	12	96	0	0	318	212	0	19.327°
109	5312.735079920	${\displaystyle C_{2}}$	0.000647299	0	0	14	93	2	0	321	214	0	19.103°
110	5413.549294192	${\displaystyle D_{6}}$	0	0	0	12	98	0	0	324	216	0	19.476°
111	5515.293214587	${\displaystyle D_{3}}$	0	0	0	12	99	0	0	327	218	0	19.255°
112	5618.044882327	${\displaystyle D_{5}}$	0	0	0	12	100	0	0	330	220	0	19.351°
113	5721.824978027	${\displaystyle D_{3}}$	0	0	0	12	101	0	0	333	222	0	18.978°
114	5826.521572163	${\displaystyle C_{2}}$	0.000149772	0	0	12	102	0	0	336	224	0	18.836°
115	5932.181285777	${\displaystyle C_{3}}$	0.000049972	0	0	12	103	0	0	339	226	0	18.458°
116	6038.815593579	${\displaystyle C_{2}}$	0.000259726	0	0	12	104	0	0	342	228	0	18.386°
117	6146.342446579	${\displaystyle C_{2}}$	0.000127609	0	0	12	105	0	0	345	230	0	18.566°
118	6254.877027790	${\displaystyle C_{2}}$	0.000332475	0	0	12	106	0	0	348	232	0	18.455°
119	6364.347317479	${\displaystyle C_{2}}$	0.000685590	0	0	12	107	0	0	351	234	0	18.336°
120	6474.756324980	${\displaystyle C_{s}}$	0.001373062	0	0	12	108	0	0	354	236	0	18.418°
121	6586.121949584	${\displaystyle C_{3}}$	0.000838863	0	0	12	109	0	0	357	238	0	18.199°
122	6698.374499261	${\displaystyle I_{h}}$	0	0	0	12	110	0	0	360	240	0	18.612°
123	6811.827228174	${\displaystyle C_{2v}}$	0.001939754	0	0	14	107	2	0	363	242	0	17.840°
124	6926.169974193	${\displaystyle D_{2}}$	0	0	0	12	112	0	0	366	244	0	18.111°
125	7041.473264023	${\displaystyle C_{2}}$	0.000088274	0	0	12	113	0	0	369	246	0	17.867°
126	7157.669224867	${\displaystyle D_{4}}$	0	0	2	16	100	8	0	372	248	0	17.920°
127	7274.819504675	${\displaystyle D_{5}}$	0	0	0	12	115	0	0	375	250	0	17.877°
128	7393.007443068	${\displaystyle C_{2}}$	0.000054132	0	0	12	116	0	0	378	252	0	17.814°
129	7512.107319268	${\displaystyle C_{2}}$	0.000030099	0	0	12	117	0	0	381	254	0	17.743°
130	7632.167378912	${\displaystyle C_{2}}$	0.000025622	0	0	12	118	0	0	384	256	0	17.683°
131	7753.205166941	${\displaystyle C_{2}}$	0.000305133	0	0	12	119	0	0	387	258	0	17.511°
132	7875.045342797	${\displaystyle I}$	0	0	0	12	120	0	0	390	260	0	17.958°
133	7998.179212898	${\displaystyle C_{3}}$	0.000591438	0	0	12	121	0	0	393	262	0	17.133°
134	8122.089721194	${\displaystyle C_{2}}$	0.000470268	0	0	12	122	0	0	396	264	0	17.214°
135	8246.909486992	${\displaystyle D_{3}}$	0	0	0	12	123	0	0	399	266	0	17.431°
136	8372.743302539	${\displaystyle T}$	0	0	0	12	124	0	0	402	268	0	17.485°
137	8499.534494782	${\displaystyle D_{5}}$	0	0	0	12	125	0	0	405	270	0	17.560°
138	8627.406389880	${\displaystyle C_{2}}$	0.000473576	0	0	12	126	0	0	408	272	0	16.924°
139	8756.227056057	${\displaystyle C_{2}}$	0.000404228	0	0	12	127	0	0	411	274	0	16.673°
140	8885.980609041	${\displaystyle C_{1}}$	0.000630351	0	0	13	126	1	0	414	276	0	16.773°
141	9016.615349190	${\displaystyle C_{2v}}$	0.000376365	0	0	14	126	0	1	417	278	0	16.962°
142	9148.271579993	${\displaystyle C_{2}}$	0.000550138	0	0	12	130	0	0	420	280	0	16.840°
143	9280.839851192	${\displaystyle C_{2}}$	0.000255449	0	0	12	131	0	0	423	282	0	16.782°
144	9414.371794460	${\displaystyle D_{2}}$	0	0	0	12	132	0	0	426	284	0	16.953°
145	9548.928837232	${\displaystyle C_{s}}$	0.000094938	0	0	12	133	0	0	429	286	0	16.841°
146	9684.381825575	${\displaystyle D_{2}}$	0	0	0	12	134	0	0	432	288	0	16.905°
147	9820.932378373	${\displaystyle C_{2}}$	0.000636651	0	0	12	135	0	0	435	290	0	16.458°
148	9958.406004270	${\displaystyle C_{2}}$	0.000203701	0	0	12	136	0	0	438	292	0	16.627°
149	10096.859907397	${\displaystyle C_{1}}$	0.000638186	0	0	14	133	2	0	441	294	0	16.344°
150	10236.196436701	${\displaystyle T}$	0	0	0	12	138	0	0	444	296	0	16.405°
151	10376.571469275	${\displaystyle C_{2}}$	0.000153836	0	0	12	139	0	0	447	298	0	16.163°
152	10517.867592878	${\displaystyle D_{2}}$	0	0	0	12	140	0	0	450	300	0	16.117°
153	10660.082748237	${\displaystyle D_{3}}$	0	0	0	12	141	0	0	453	302	0	16.390°
154	10803.372421141	${\displaystyle C_{2}}$	0.000735800	0	0	12	142	0	0	456	304	0	16.078°
155	10947.574692279	${\displaystyle C_{2}}$	0.000603670	0	0	12	143	0	0	459	306	0	15.990°
156	11092.798311456	${\displaystyle C_{2}}$	0.000508534	0	0	12	144	0	0	462	308	0	15.822°
157	11238.903041156	${\displaystyle C_{2}}$	0.000357679	0	0	12	145	0	0	465	310	0	15.948°
158	11385.990186197	${\displaystyle C_{2}}$	0.000921918	0	0	12	146	0	0	468	312	0	15.987°
159	11534.023960956	${\displaystyle C_{2}}$	0.000381457	0	0	12	147	0	0	471	314	0	15.960°
160	11683.054805549	${\displaystyle D_{2}}$	0	0	0	12	148	0	0	474	316	0	15.961°
161	11833.084739465	${\displaystyle C_{2}}$	0.000056447	0	0	12	149	0	0	477	318	0	15.810°
162	11984.050335814	${\displaystyle D_{3}}$	0	0	0	12	150	0	0	480	320	0	15.813°
163	12136.013053220	${\displaystyle C_{2}}$	0.000120798	0	0	12	151	0	0	483	322	0	15.675°
164	12288.930105320	${\displaystyle D_{2}}$	0	0	0	12	152	0	0	486	324	0	15.655°
165	12442.804451373	${\displaystyle C_{2}}$	0.000091119	0	0	12	153	0	0	489	326	0	15.651°
166	12597.649071323	${\displaystyle D_{2d}}$	0	0	0	16	146	4	0	492	328	0	15.607°
167	12753.469429750	${\displaystyle C_{2}}$	0.000097382	0	0	12	155	0	0	495	330	0	15.600°
168	12910.212672268	${\displaystyle D_{3}}$	0	0	0	12	156	0	0	498	332	0	15.655°
169	13068.006451127	${\displaystyle C_{s}}$	0.000068102	0	0	13	155	1	0	501	334	0	15.537°
170	13226.681078541	${\displaystyle D_{2d}}$	0	0	0	12	158	0	0	504	336	0	15.569°
171	13386.355930717	${\displaystyle D_{3}}$	0	0	0	12	159	0	0	507	338	0	15.497°
172	13547.018108787	${\displaystyle C_{2v}}$	0.000547291	0	0	14	156	2	0	510	340	0	15.292°
173	13708.635243034	${\displaystyle C_{s}}$	0.000286544	0	0	12	161	0	0	513	342	0	15.225°
174	13871.187092292	${\displaystyle D_{2}}$	0	0	0	12	162	0	0	516	344	0	15.366°
175	14034.781306929	${\displaystyle C_{2}}$	0.000026686	0	0	12	163	0	0	519	346	0	15.252°
176	14199.354775632	${\displaystyle C_{1}}$	0.000283978	0	0	12	164	0	0	522	348	0	15.101°
177	14364.837545298	${\displaystyle D_{5}}$	0	0	0	12	165	0	0	525	350	0	15.269°
178	14531.309552587	${\displaystyle D_{2}}$	0	0	0	12	166	0	0	528	352	0	15.145°
179	14698.754594220	${\displaystyle C_{1}}$	0.000125113	0	0	13	165	1	0	531	354	0	14.968°
180	14867.099927525	${\displaystyle D_{2}}$	0	0	0	12	168	0	0	534	356	0	15.067°
181	15036.467239769	${\displaystyle C_{2}}$	0.000304193	0	0	12	169	0	0	537	358	0	15.002°
182	15206.730610906	${\displaystyle D_{5}}$	0	0	0	12	170	0	0	540	360	0	15.155°
183	15378.166571028	${\displaystyle C_{1}}$	0.000467899	0	0	12	171	0	0	543	362	0	14.747°
184	15550.421450311	${\displaystyle T}$	0	0	0	12	172	0	0	546	364	0	14.932°
185	15723.720074072	${\displaystyle C_{2}}$	0.000389762	0	0	12	173	0	0	549	366	0	14.775°
186	15897.897437048	${\displaystyle C_{1}}$	0.000389762	0	0	12	174	0	0	552	368	0	14.739°
187	16072.975186320	${\displaystyle D_{5}}$	0	0	0	12	175	0	0	555	370	0	14.848°
188	16249.222678879	${\displaystyle D_{2}}$	0	0	0	12	176	0	0	558	372	0	14.740°
189	16426.371938862	${\displaystyle C_{2}}$	0.000020732	0	0	12	177	0	0	561	374	0	14.671°
190	16604.428338501	${\displaystyle C_{3}}$	0.000586804	0	0	12	178	0	0	564	376	0	14.501°
191	16783.452219362	${\displaystyle C_{1}}$	0.001129202	0	0	13	177	1	0	567	378	0	14.195°
192	16963.338386460	${\displaystyle I}$	0	0	0	12	180	0	0	570	380	0	14.819°
193	17144.564740880	${\displaystyle C_{2}}$	0.000985192	0	0	12	181	0	0	573	382	0	14.144°
194	17326.616136471	${\displaystyle C_{1}}$	0.000322358	0	0	12	182	0	0	576	384	0	14.350°
195	17509.489303930	${\displaystyle D_{3}}$	0	0	0	12	183	0	0	579	386	0	14.375°
196	17693.460548082	${\displaystyle C_{2}}$	0.000315907	0	0	12	184	0	0	582	388	0	14.251°
197	17878.340162571	${\displaystyle D_{5}}$	0	0	0	12	185	0	0	585	390	0	14.147°
198	18064.262177195	${\displaystyle C_{2}}$	0.000011149	0	0	12	186	0	0	588	392	0	14.237°
199	18251.082495640	${\displaystyle C_{1}}$	0.000534779	0	0	12	187	0	0	591	394	0	14.153°
200	18438.842717530	${\displaystyle D_{2}}$	0	0	0	12	188	0	0	594	396	0	14.222°
201	18627.591226244	${\displaystyle C_{1}}$	0.001048859	0	0	13	187	1	0	597	398	0	13.830°
202	18817.204718262	${\displaystyle D_{5}}$	0	0	0	12	190	0	0	600	400	0	14.189°
203	19007.981204580	${\displaystyle C_{s}}$	0.000600343	0	0	12	191	0	0	603	402	0	13.977°
204	19199.540775603	${\displaystyle T_{h}}$	0	0	0	12	192	0	0	606	404	0	14.291°
212	20768.053085964	${\displaystyle I}$	0	0	0	12	200	0	0	630	420	0	14.118°
214	21169.910410375	${\displaystyle T}$	0	0	0	12	202	0	0	636	424	0	13.771°
216	21575.596377869	${\displaystyle D_{3}}$	0	0	0	12	204	0	0	642	428	0	13.735°
217	21779.856080418	${\displaystyle D_{5}}$	0	0	0	12	205	0	0	645	430	0	13.902°
232	24961.252318934	${\displaystyle T}$	0	0	0	12	220	0	0	690	460	0	13.260°
255	30264.424251281	${\displaystyle D_{3}}$	0	0	0	12	243	0	0	759	506	0	12.565°
256	30506.687515847	${\displaystyle T}$	0	0	0	12	244	0	0	762	508	0	12.572°
257	30749.941417346	${\displaystyle D_{5}}$	0	0	0	12	245	0	0	765	510	0	12.672°
272	34515.193292681	${\displaystyle I_{h}}$	0	0	0	12	260	0	0	810	540	0	12.335°
282	37147.294418462	${\displaystyle I}$	0	0	0	12	270	0	0	840	560	0	12.166°
292	39877.008012909	${\displaystyle D_{5}}$	0	0	0	12	280	0	0	870	580	0	11.857°
306	43862.569780797	${\displaystyle T_{h}}$	0	0	0	12	294	0	0	912	608	0	11.628°
312	45629.313804002	${\displaystyle C_{2}}$	0.000306163	0	0	12	300	0	0	930	620	0	11.299°
315	46525.825643432	${\displaystyle D_{3}}$	0	0	0	12	303	0	0	939	626	0	11.337°
317	47128.310344520	${\displaystyle D_{5}}$	0	0	0	12	305	0	0	945	630	0	11.423°
318	47431.056020043	${\displaystyle D_{3}}$	0	0	0	12	306	0	0	948	632	0	11.219°
334	52407.728127822	${\displaystyle T}$	0	0	0	12	322	0	0	996	664	0	11.058°
348	56967.472454334	${\displaystyle T_{h}}$	0	0	0	12	336	0	0	1038	692	0	10.721°
357	59999.922939598	${\displaystyle D_{5}}$	0	0	0	12	345	0	0	1065	710	0	10.728°
358	60341.830924588	${\displaystyle T}$	0	0	0	12	346	0	0	1068	712	0	10.647°
372	65230.027122557	${\displaystyle I}$	0	0	0	12	360	0	0	1110	740	0	10.531°
382	68839.426839215	${\displaystyle D_{5}}$	0	0	0	12	370	0	0	1140	760	0	10.379°
390	71797.035335953	${\displaystyle T_{h}}$	0	0	0	12	378	0	0	1164	776	0	10.222°
392	72546.258370889	${\displaystyle I}$	0	0	0	12	380	0	0	1170	780	0	10.278°
400	75582.448512213	${\displaystyle T}$	0	0	0	12	388	0	0	1194	796	0	10.068°
402	76351.192432673	${\displaystyle D_{5}}$	0	0	0	12	390	0	0	1200	800	0	10.099°
432	88353.709681956	${\displaystyle D_{3}}$	0	0	0	24	396	12	0	1290	860	0	9.556°
448	95115.546986209	${\displaystyle T}$	0	0	0	24	412	12	0	1338	892	0	9.322°
460	100351.763108673	${\displaystyle T}$	0	0	0	24	424	12	0	1374	916	0	9.297°
468	103920.871715127	${\displaystyle S_{6}}$	0	0	0	24	432	12	0	1398	932	0	9.120°
470	104822.886324279	${\displaystyle S_{6}}$	0	0	0	24	434	12	0	1404	936	0	9.059°

1. Thomson, Joseph John (March 1904). “On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure”. Philosophical Magazine. Series 6 7 (39): 237–265. doi:10.1080/14786440409463107. オリジナルの13 December 2013時点におけるアーカイブ。. https://web.archive.org/web/20131213172104/http://www.cond-mat.physik.uni-mainz.de/~oettel/ws10/thomson_PhilMag_7_237_1904.pdf.
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6. Andreev, N.N. (1996). “An extremal property of the icosahedron”. East J. Approximation 2 (4): 459–462. MR1426716, Zbl 0877.51021
7. Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
8. Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
9. Y. Levin and J. J. Arenzon, ``Why charges go to the Surface: A generalized Thomson Problem Europhys. Lett. Vol. 63 p. 415 (2003)
10. Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
11. LaFave Jr, Tim (December 2013). “Correspondences between the classical electrostatic Thomson problem and atomic electronic structure”. Journal of Electrostatics 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001. オリジナルの22 February 2014時点におけるアーカイブ。. https://web.archive.org/web/20140222100807/http://www.pagesofmind.com/FullTextPubs/La13-LaFave-2013-Correspondences-between-the-Thomson-Problem-and-Atomic-Structure.pdf 2014年2月10日閲覧。.
12. Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.

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